A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — a parabola

Detailed Solution

Let the centre be $(h, k)$ with $k > 0$ ; touching the X-axis makes the radius $k$ .

Touching the other circle: $h^{2} + (k - 3)^{2} = k + 2$ (externally) or $∣ k - 2 ∣$ (internally).

Squaring the external case: $h^{2} - 6 k + 9 = 4 k + 4$ , i.e. $x^{2} = 10 y - 5$ , a parabola (the internal case gives $x^{2} = 2 y - 5$ , also a parabola).

✓ Check

(0, \frac{1}{2})

is on

x^{2} = 10 y - 5

; radius

\frac{1}{2}

, distance to (0, 3) is

\frac{5}{2} = \frac{1}{2} + 2

⚠ Common trap

k + 2

, which changes with

k

Quick tip

Centre $(h, k)$ , radius $k$ : $h^{2} + (k - 3)^{2} = (k + 2)^{2} \Rightarrow h^{2} = 10 k - 5$ , a parabola

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Circle — Preparation

This question tests your understanding of Circle, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

In coordinate geometry, a circle is the set of points at a fixed distance (the radius) from a fixed point (the centre). This topic covers the standard equation (x - h)^2 + (y - k)^2 = r^2, the general equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f), the diameter form, and parametric equations. Exam questions also test the position of a point relative to a circle, the condition for a line to be a tangent, equations of tangents and normals, the length of a tangent from an external point, and the intersection of two circles. The fastest approach is often to compare the distance from the centre to a line with the radius: equal means tangent, smaller means the line cuts the circle. Practise the circle questions below, each with a step-by-step solution.

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A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is

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